Digital integration is a common DSP operation used in, for example, infinite-impulse response (IIR) filters, cascaded integrator-comb (CIC) filters and statistical signal parameter estimation. In such applications, the order of the integration integral is generally unity.
FIG. 1 is a schematic block diagram illustrating a prior art implementation of a digital integrator 10 with unity order of integration. In FIG. 1, a summer 12 sums the input signal xi on line 14 with feedback signal yi-1 on line 16 to produce the output signal yi on line 18. A delay register z−1 20 delays signal yi on line 18 by one sample delay producing the signal yi-1 on line 22. The following equation (EQ. 1) gives a mathematical description for the computation performed by the digital integrator circuit illustrated in FIG. 1:
      y    i    =            ∑              i        =        0            n        ⁢          (                        x          i                +                  y                      i            -            1                              )      
Alternatively, the following equation (EQ. 2) is the z-transform description of the digital integrator circuit illustrated in FIG. 1:
      y    ⁡          (      z      )        =            1              1        -                  z                      -            1                                ⁢          x      ⁡              (        z        )            
This prior art approach is restricted to performing digital integration where the order of the integration integral is unity.
Digital integration using non-integer or fractional-order integrals is a mathematical operation defined by the mathematical field of fractional calculus that computes integrals having non-integer order. Fractional calculus for the continuous domain was first described by Bernhard Riemann and Joseph Liouville in 1832 with what is now called the Riemann-Liouville integral. There are many applications for fractional integration. Fractional order control, a field of control theory, relies on fractional integration of measured parameters to produce control values for control actuators. In signal pulse detection, digital fractional integrators have application in the efficient detection of received signal pulses with unknown pulse heights and pulse rise times in the presence of noise.
The Grunwald-Letnikov differintegral, given by the following equation (EQ. 3), is a mathematical definition of the fractional integral of a continuous function ƒ(x) for fractional integration parameter α.
            D              -        α              ⁢          f      ⁡              (        x        )              =            lim              n        ->        ∞              ⁢                            (                      n                          x              -              α                                )                α            ⁢                        ∑                      i            =            0                    n                ⁢                                            Γ              ⁡                              (                                  α                  +                  i                                )                                                                    i                !                            ⁢                              Γ                ⁡                                  (                  α                  )                                                              ⁢                      f            ⁡                          (                              x                -                                  i                  ⁡                                      (                                                                  x                        -                        α                                            n                                        )                                                              )                                          
It would be desirable to be able to implement the Grunwald-Letnikov differintegral approach of EQ. 3 as well as similar approaches in a hardware, firmware, and/or software implementation for use in, for example, digital signal processing of received signals.